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A Curious Bijection on Natural Numbers
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B. J. Venkatachala

MO Cell, HBCSE(TIFR)

Department of Mathematics

Indian Institute of Science

Bangalore-560012

India

**Abstract:**

We give a greedy algorithm for describing an enumeration of the set of all
natural numbers such that the sum of the first *n* terms of the
sequence is divisible by *n* for each natural number *n*. We show
that this leads to a bijection *f* of the set of all natural
numbers onto itself that has some nice properties. We also show
that the average function of the first *n* terms of the
sequence satisfies a functional equation which completely describes
all the properties of the function *f*.
In particular, *f* turns out to
be an *involution* on the set of all natural numbers.

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(Concerned with sequence
A019444.)

Received June 16 2009;
revised version received November 11 2009.
Published in *Journal of Integer Sequences*, November 16 2009.

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